Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence. And let $L^+$ be an extended real number (i.e. $L^+\in\mathbb{R}^*$). Then TFAE: (1) $L^+$ = $\displaystyle\inf_{n\in\mathbb{N}}\sup_{k{\geq}n}(a_k)$ = $\displaystyle\lim_{n\to\infty}\sup_{k{\geq}n}(a_k)$ (2) $L^+$ = $\sup\{p\in\mathbb{R}^*:p$ is a cluster point of $(a_n)_{n\in\mathbb{N}}\}$ (3) $L^+$ is a (or the unique) number in $\mathbb{R}^*$ satisfying: (i) For all $\epsilon\gt 0$ there exists $N\in\mathbb{N}$ such that […]

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + 1^3$ Here’s the argument: Assume $p \ge 5$ with $x,y,p,n$ positive integers and $p^n = x^3 + […]

Background I am trying to prove the following theorem. Let $f:[a,b]\to\mathbb{R}$ be a bounded function. If $c\in(a,b)$ then show that $f:[a,b]\to\mathbb{R}$ is Riemann Integrable on $[a,b]$ if $f$ is Riemann Integrable on both $[a,c]$ and $[c,b]$. Notation We use the following notations to simplify our discussion. The collection of all partitions on $[a,b]$ for the […]

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel’s differential equation is $$x^2y^{\prime\prime}+xy^{\prime} + (x^2 – p^2)y=0\tag{1}$$ where $(1)$ has a first solution given by $$\fbox{$J_p(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+p)}\left(\frac{x}{2}\right)^{2n+p}$}\tag{2}$$ and a second solution given by $$\fbox{$J_{-p}(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1-p)}\left(\frac{x}{2}\right)^{2n-p}$}\tag{3}$$ where $J_p(x)$ is called […]

Prove with induction on $n$ that \begin{align*} \Bigl(1+ \frac{1}{n}\Bigr)^n < n \end{align*} for natural numbers $n \geq 3$. Attempt at proof: Basic step. This can be verified easily. Induction step. Suppose the assertion holds for $n >3$, then we now prove it for $n+1$. We want to prove that \begin{align*} \big( 1+ \frac{1}{n+1})^{n+1} < n+1. […]

Definition 2.15. A group action of $G$ on $X$ is called faithful (or effective) if different elements of $G$ act on $X$ in different ways: when $g_1\neq g_2$ in G, there is an $x\in X$ such that $g_1\cdot x \neq g_2\cdot x$. Example 2.17. The action of $G$ on itself by conjugation is faithful if […]

Is this proof acceptable ? Lemma Let $M_p=2^p-1$ with $p$ prime and $p>2$ , thus If $M_p$ is prime then $3^{\frac{M_p-1}{2}} \equiv -1 \pmod {M_p}$ Proof Let $M_p$ be a prime , then according to Euler’s criterion : $3^{\frac{M_p-1}{2}} \equiv \left(\frac{3}{M_p}\right) \pmod {M_p}$ , where $\left(\frac{3}{M_p}\right)$ denotes Legendre symbol . If $M_p$ is prime then […]

Let $f$ is continuous and strictly monotonic function on $[a,b]$. Prove that $$f([a,b])=[f(a),f(b)].$$ Proof: We know that $[a,b]$ is a connected set in $\mathbb{R}$ then $f([a,b])$ is connected since $f$ is continuous. Also $f(a),f(b)\in f([a,b])$. Hence $[f(a),f(b)]\subseteq f([a,b]).$ Let $y\in f([a,b])$ then $y=f(x)$ for some $x\in[a,b]$. Since $x\in[a,b]$ then $f(a)\leqslant y\leqslant f(b)$. Hence $f([a,b])\subseteq [f(a),f(b)]$ […]

Working a problem stated in Enderton, but stated better and apparently stronger in Soare. All citations hereon are for Soare (1987). Would appreciate help on the proof. I know there has to be a more elegant proof than what I have attempted, assuming my proof is correct (it may not be). The problem is stated […]

Let $K/F$ be a Galois extension with $G=\text{Gal}(K/F)$. $\mathcal{I}=\lbrace E\text{ }|\text{ }E/F \text{ is Galois and }[E:F]<\infty\rbrace$ $\mathcal{N}=\lbrace N\text{ }|\text{ } N=\text{Gal}(K/E) \text{ for some } E\in\mathcal{I}\rbrace$ The Krull topology over $G$ is defined as follows. The set $B=\lbrace \sigma_iN_i\text{ }|\text{ }\sigma_i\in G \text{ and }N_i\in\mathcal{N}\rbrace$ is the base for the Krull topology. I am […]

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